All of the 3rd grade teachers and students from Gardner Bullis went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$3.50$ each for students, and the group paid $$38.00$ in total. The next month, the same group visited a science museum where the tickets cost $$34.00$ each for teachers and $$10.00$ each for students, and the group paid $$128.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+3.5y = 38}$ ${34x+10y = 128}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-14y = -152}$ ${34x+10y = 128}$ Add the top and bottom equations together. $ -4y = -24 $ $ y = \dfrac{-24}{-4}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {8.5x+3.5y = 38}$ to find $x$ ${8.5x + 3.5}{(6)}{= 38}$ $8.5x+21 = 38$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 6}$ into $ {34x+10y = 128}$ and get the same answer for $x$ ${34x + 10}{(6)}{= 128}$ ${x = 2}$ There were $2$ teachers and $6$ students on the field trips.